Crack Paralleling an Interface Between Dissimilar Materials
J. W. Hutchinson M. E. Mear1 J. R. Rice Division of Applied Sciences, Harvard University, Cambridge, MA 02138
1
Crack Paralleling an Interface Between Dissimilar Materials A crack paralleling a bonded plane interface between two dissimilar isotropic elastic solids is considered. When the distance of the crack from the interface is small compared to the crack length itself and to other length scales characterizing the geometry, a simple universal relation exists between the Mode I and Mode II stress intensity factors and the complex stress intensity factor associated with the corresponding problem for the crack lying on the interface. In other words, if the influence of external loading and geometry on the interface crack is known, then this information can immediately be used to generate the stress intensity factors for the sub-interface crack. Conditions for cracks to propagate near and parallel to, but not along, an interface are derived.
Introduction
INTERFACE CRACK
Bonded interfaces between dissimilar elastic materials often separate by cracking, as would be expected if the toughness of the interface is low compared to that of the abutting materials. In some instances cracking is observed to occur approximately parallel to the interface but with the crack lying entirely within one of the two materials. The aim of this paper is to analyze subinterface cracks which parallel the interface and to examine conditions under which they might be expected. The mathematical problem which is analyzed is introduced in Fig. 1. Each material is taken to be isotropic and linearly elastic. The interface lies along the xx axis with material #1 lying above and #2 below. Plane strain deformations are considered. Attention will be restricted to subinterface cracks which lie below the interface at a distance h which is small compared to the length of the crack L and to all other relevant geometric length quantities in the problem. As indicated in Fig. 1, we will consider the asymptotic problem for the semiinfinite subinterface crack. The remote field in the asymptotic problem is prescribed to be the near-tip field of the interface crack problem (everywhere but in material #2 between the crack and the interface). That is, the solution to the subinterface crack problem at any point a fixed distance from the tip approaches the solution to the corresponding interface crack problem as h~0+ with L fixed. Thus, at distances from the tip which are large compared to h and small compared to L, the near-tip field of the interface crack problem pertains. Posing the problem in this manner permits us to develop a universal relation between the Mode I and II stress intensity factors of the subinterface crack and the corresponding "complex" stress intensity factor of the interface crack. This relation is Currently, Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, Austin, Texas. Contributed by the Applied Mechanics Division for publication in the JOURNAL OF APPLIED MECHANICS.
Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York, N.Y. 10017, and will be accepted until two months after final publication of the paper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by ASME Applied Mechanics Division, May 11, 1987.
SUB-INTERFACE CRACK
SEMI-INFINITE SUB-INTERFACE CRACK
Asymptotic remote field merges with near-tip field of interface crack problem Fig. 1 Relation of asymptotic subinterface crack problem to interface crack problem
otherwise independent of loading, crack length, and external geometry. With the universal relation in hand, we examine conditions under which propagation of a parallel sub-interface crack should be expected. When conditions do favor such cracks, the analysis predicts the separation distance from the interface. 2
Formulation and Solution The singular near-tip field of the interface crack problem
828/Vol. 54, DECEMBER 1987
Transactions of the ASME
Copyright © 1987 by ASME Downloaded 21 May 2012 to 128.103.149.52. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Table 1
X
-.8
-.6
-.4
-.2
-.1
V a l u e s o f («, /S)
0
.1
.2
.4
.6
.8
.0810
. -.4
-.1202
-.0801
-.0467
-.0186
-.0060
.0058
.0168
.0273
.0468
.0653
-.2
-.1814
-.1162
-.0708
-.0351
-.0197
-.0056
.0075
.0197
.0419
.0618
.0798
-.1
-.2057
-.1281
-.0764
-.0368
-.0199
-.0046
.0096
.0227
.0465
.0675
.0865
-.05
-.2167
-.1328
-.0779
-.0363
-.0187
-.0027
.0120
.0256
.0501
.0718
.0912
-.02
-.2229
-.1354
-.0785
-.0356
-.0176
-.0012
.0139
.0277
.0527
.0748
.0946
0
-.2270
-.1369
-.0787
-.0350
-.0167
0
.0153
.0293
.0547
.0770
.0970
.02
-.2309
-.1384
-.0788
-.0343
-.0156
.0013
.0168
.0311
,0567
.0793
.0995
.05
-.2366
-.1403
-.0187
-.0330
-.0138
.0035
.0193
.0339
.0601
.0830
.1035
.]
-2456
-.1431
-.0780
-.0301
-.0101
.0079
.0243
.0393
.0663
.0900
.1110
.2
-.2620
-.1468
-.0744
-.0219
-.0003
.0191
.0367
.0528
.0815
.1065
.1287
.4
-2902
-.1449
-.0566
.0055
.0307
.0531
.0733
.0917
.1242
.1522
.1769
.
(England, 1965; Erdogan, 1965; Rice and Sih, 1965) gives rise to tractions directly ahead of the tip (6 = 0) given by
with universal (complex) angular dependence cfaj3(0) for a given material pair. The remote crack face displacements ap<j21 + ion=K(2w)-mrk (2.1) proach equation (2.3). The only length quantity in the semiinfinite sub-interface crack problem is h. From dimensional where K=Kx+iK2 is the complex stress intensity factor, considerations and by linearity it follows that / = V ( - 1 ) , and Ki+^^cKh't+dKh-" (2.9) 1 . f G^G^-Av^ In t = (2.2) where c and d are dimensionless complex constants depending 2TT L G2 + G,(3-4y 2 ) only on dimensionless combinations of the moduli of the where G is the shear modulus and v is Poisson's ratio. Here materials. The depth of the crack below the interface must apK= (kx + ik2yfir cosh 7re where kx + ik2 is the complex intensi- pear as the factor hk to combine with L~k in equation (2.4) as ty factor as originally introduced by Rice and Sih (1965). The the dimensionless term (h/L)K. Vir is standard in converting the lower case k's of that period By considering a unit advance of the semi-infinite crack, to K's; we include the factor cosh ire so that the magnitude of one concludes by an energy argument, or equivalently by apthe traction vector on the interface is given by plication of the /integral, that the energy release-rate given by X ( ff22 + a\2) = I K I / V 2 7r r , a n a l o g o u s l y t o equation (2.7) must be equal to that given in equation (2.5). the homogeneous material case. The associated crack face That is displacements a distance r behind the tip are given by K2+K2l=q2KK (2.10) Ki-^/G.+a-i^/Gj] 1 of the elastic moduli. A further simplification is achieved when use is made of Dundurs' (1969) observation that for problems of this class the moduli dependence can be expressed in terms of just two (rather than three) special nondimensional combinations. In plane strain, Dundurs' parameters are
„
/2
a22 + ian^(Kl+iKn){2itr)^
1~"2
(2.7) -](*?+*?,) 2G, As discussed earlier, the remote stresses in the semi-infinite subsurface crack problem are required to approach (for all 6 but 6 = if) the characteristic Williams singular field of the interface crack, which can be written as
Journal of Applied Mechanics
G . d - ^ + G^l-K,)
(2.8)
(2.13)
and 1 G,(l-2^)-G2(l-2v1) 0 , ( 1 - ^ + 0,(1-^)
0= 2
(2.6)
where Kx and Ku are the standard Mode I and Mode II stress intensity factors. The energy release-rate is
Khk (2.18) feature of the solution for arbitrary small values of h/L. If the collection of terms <j> + e ln(h/L) is small, as might easily be the Note that AT, =Kl and Kn =K2 when a and (3 both vanish. ease judging from the systems listed in Table 2, then the stress intensity factors are well approximated by 3 Applications and Implications Moduli and values of a, /3, e, and / are presented for six representative material combinations in Table 2. The shear modulus and Poisson's ratio listed for each material are polycrystalline values derived from Simmons and Wang (1971). The values for the cubic materials are the average of 830/Vol. 54, DECEMBER 1987
Kj + iKn = qLkK =q\K\ eh
(3.6)
For example, in the case of the concentrated wedge force (3.1) Kl+iKll=q(P+iQ)(t!L/2Y'n
(3.7)
assuming e itself is small. Apart from the factor q, this is just Transactions of the ASME
Downloaded 21 May 2012 to 128.103.149.52. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
the classical result when no material discontinuity occurs. Similarly, the result for the finite crack (3.2) becomes
the other systems may occur for other geometries and loadings. The discussion and the analysis given above assume that 1/2 K, + iKu=q(o% + /(rr2)(7rZ,/2) (3.8) contact between the crack faces of the subinterface crack does which is again the classical result multiplied by q. In other not occur. In applications where the near tip conditions of the words, in these examples when e and <j> + e ln(h/L) are both subinterface crack is in pure Mode I and where h/L is not small, the ratio of the Mode II to Mode I stress intensity fac- ludicrously small, it is unlikely that contact of the crack faces tors is the same as in the corresponding classical problem but will be an issue. If the corresponding interface crack problem the energy release-rate is that of the interface crack as reflected does indicate contact well away from the tip, at distances as large as h or greater, then the possibility of contact in the by the factor q. subinterface crack problem should certainly be checked. SoluNow consider situations in fatigue, stress corrosion or under tions to (3.12) for h/L are only physically meaningful when monotonic loading when the crack will tend to advance in its h/L is not so small that contact will certainly occur or, what is own plane approximately parallel to the interface. Assuming more likely, that h is not so small that the crack lies so close to the fracture properties of material #2 are homogeneous along the interface that the material at the tip has properties which with its moduli, the crack will only advance in its plane if are affected by the existence of the interface. Kn=0. If Kn > 0 it will tend to deflect downward away from the interface, while if Kn < 0 it will tend to grow upward. By equation (3.5), the condition for the crack to advance parallel Acknowledgment to the interface in pure Mode I is This work was supported in part by the Office of Naval Research under Contract ONR-N00014-86-K-0753sin[7 + 0 + e/n(/)/L)] = 0 (3.9) Subagreement VB38639-0 from the University of California, or Santa Barbara, California, and by the Division of Applied y + <j> + eln(h/L)=2irn; « = 0,±1, ... (3.10) Sciences, Harvard University. with the associated Mode I intensity Kl=q\K\
References (3.11)
Values of h/L from equations (3.10) are h/L = exp[(2irn-y-)/e];
« = 0,±1, ...
(3.12)
but only those values (if any) will be physically meaningful which are small compared to unity but not so small that the parts of the crack faces make contact, as will be discussed below. The crack length L increases as the crack advances and thus h cannot remain strictly constant. However, if h at the tip satisfies equations (3.12) approximately as L increases the slope dh/dL of the crack, the path will be small (and equal to the value given by equations (3.12)), with the crack thus nearly paralleling the interface when h/L is small. As an illustration, consider the symmetric wedge loading (Q = 0) of the geometry in Fig. 2(a). By equations (3.1) and (3.3), 7 = 0. For the material systems listed in Table 2, the largest magnitude of e is 0.04, and it is readily seen that the only physically meaningful solution from equations (3.12), if any, is that associated with n = 0, i.e., h/L = exp[-<j>/e]
(3.13)
Of the systems in Table 2, only Cu/Si, Si/Cu, Ni/MgO, and MgO/Ni have positive values of $/e and might therefore propagate a subinterface crack of the kind envisioned here for this particular geometry and loading. For Cu/Si, h/L = 0.26; while for Si/Cu, h/L = 0.32. The accuracy of these estimates may be somewhat questionable since they probably lie outside the range of h/L where the asymptotic analysis is accurate. For Ni/MgO, h/L = 0.009 and for MgO/Ni, h/L = 0.011, and these estimates should be accurate. Evidently the crack could satisfy a Kn = 0 criterion by propagating near the interface in either phase. We do not investigate here the configurational stability of those paths but expect, following Cotterell and Rice (1980), that only a path with a negative crack-parallel nonsingular stress term at the tip is stable. The conclusions for a finite crack paralleling the interface in Fig. 2(b) under remote tensile loading ( ^ = 0) are similar. Now, 7 = 2e and h/L = exp[-2- is comparable.
Transactions of the ASME
Downloaded 21 May 2012 to 128.103.149.52. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
1
Crack Paralleling an Interface Between Dissimilar Materials A crack paralleling a bonded plane interface between two dissimilar isotropic elastic solids is considered. When the distance of the crack from the interface is small compared to the crack length itself and to other length scales characterizing the geometry, a simple universal relation exists between the Mode I and Mode II stress intensity factors and the complex stress intensity factor associated with the corresponding problem for the crack lying on the interface. In other words, if the influence of external loading and geometry on the interface crack is known, then this information can immediately be used to generate the stress intensity factors for the sub-interface crack. Conditions for cracks to propagate near and parallel to, but not along, an interface are derived.
Introduction
INTERFACE CRACK
Bonded interfaces between dissimilar elastic materials often separate by cracking, as would be expected if the toughness of the interface is low compared to that of the abutting materials. In some instances cracking is observed to occur approximately parallel to the interface but with the crack lying entirely within one of the two materials. The aim of this paper is to analyze subinterface cracks which parallel the interface and to examine conditions under which they might be expected. The mathematical problem which is analyzed is introduced in Fig. 1. Each material is taken to be isotropic and linearly elastic. The interface lies along the xx axis with material #1 lying above and #2 below. Plane strain deformations are considered. Attention will be restricted to subinterface cracks which lie below the interface at a distance h which is small compared to the length of the crack L and to all other relevant geometric length quantities in the problem. As indicated in Fig. 1, we will consider the asymptotic problem for the semiinfinite subinterface crack. The remote field in the asymptotic problem is prescribed to be the near-tip field of the interface crack problem (everywhere but in material #2 between the crack and the interface). That is, the solution to the subinterface crack problem at any point a fixed distance from the tip approaches the solution to the corresponding interface crack problem as h~0+ with L fixed. Thus, at distances from the tip which are large compared to h and small compared to L, the near-tip field of the interface crack problem pertains. Posing the problem in this manner permits us to develop a universal relation between the Mode I and II stress intensity factors of the subinterface crack and the corresponding "complex" stress intensity factor of the interface crack. This relation is Currently, Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, Austin, Texas. Contributed by the Applied Mechanics Division for publication in the JOURNAL OF APPLIED MECHANICS.
Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York, N.Y. 10017, and will be accepted until two months after final publication of the paper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by ASME Applied Mechanics Division, May 11, 1987.
SUB-INTERFACE CRACK
SEMI-INFINITE SUB-INTERFACE CRACK
Asymptotic remote field merges with near-tip field of interface crack problem Fig. 1 Relation of asymptotic subinterface crack problem to interface crack problem
otherwise independent of loading, crack length, and external geometry. With the universal relation in hand, we examine conditions under which propagation of a parallel sub-interface crack should be expected. When conditions do favor such cracks, the analysis predicts the separation distance from the interface. 2
Formulation and Solution The singular near-tip field of the interface crack problem
828/Vol. 54, DECEMBER 1987
Transactions of the ASME
Copyright © 1987 by ASME Downloaded 21 May 2012 to 128.103.149.52. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
Table 1
X
-.8
-.6
-.4
-.2
-.1
V a l u e s o f («, /S)
0
.1
.2
.4
.6
.8
.0810
. -.4
-.1202
-.0801
-.0467
-.0186
-.0060
.0058
.0168
.0273
.0468
.0653
-.2
-.1814
-.1162
-.0708
-.0351
-.0197
-.0056
.0075
.0197
.0419
.0618
.0798
-.1
-.2057
-.1281
-.0764
-.0368
-.0199
-.0046
.0096
.0227
.0465
.0675
.0865
-.05
-.2167
-.1328
-.0779
-.0363
-.0187
-.0027
.0120
.0256
.0501
.0718
.0912
-.02
-.2229
-.1354
-.0785
-.0356
-.0176
-.0012
.0139
.0277
.0527
.0748
.0946
0
-.2270
-.1369
-.0787
-.0350
-.0167
0
.0153
.0293
.0547
.0770
.0970
.02
-.2309
-.1384
-.0788
-.0343
-.0156
.0013
.0168
.0311
,0567
.0793
.0995
.05
-.2366
-.1403
-.0187
-.0330
-.0138
.0035
.0193
.0339
.0601
.0830
.1035
.]
-2456
-.1431
-.0780
-.0301
-.0101
.0079
.0243
.0393
.0663
.0900
.1110
.2
-.2620
-.1468
-.0744
-.0219
-.0003
.0191
.0367
.0528
.0815
.1065
.1287
.4
-2902
-.1449
-.0566
.0055
.0307
.0531
.0733
.0917
.1242
.1522
.1769
.
(England, 1965; Erdogan, 1965; Rice and Sih, 1965) gives rise to tractions directly ahead of the tip (6 = 0) given by
with universal (complex) angular dependence cfaj3(0) for a given material pair. The remote crack face displacements ap<j21 + ion=K(2w)-mrk (2.1) proach equation (2.3). The only length quantity in the semiinfinite sub-interface crack problem is h. From dimensional where K=Kx+iK2 is the complex stress intensity factor, considerations and by linearity it follows that / = V ( - 1 ) , and Ki+^^cKh't+dKh-" (2.9) 1 . f G^G^-Av^ In t = (2.2) where c and d are dimensionless complex constants depending 2TT L G2 + G,(3-4y 2 ) only on dimensionless combinations of the moduli of the where G is the shear modulus and v is Poisson's ratio. Here materials. The depth of the crack below the interface must apK= (kx + ik2yfir cosh 7re where kx + ik2 is the complex intensi- pear as the factor hk to combine with L~k in equation (2.4) as ty factor as originally introduced by Rice and Sih (1965). The the dimensionless term (h/L)K. Vir is standard in converting the lower case k's of that period By considering a unit advance of the semi-infinite crack, to K's; we include the factor cosh ire so that the magnitude of one concludes by an energy argument, or equivalently by apthe traction vector on the interface is given by plication of the /integral, that the energy release-rate given by X ( ff22 + a\2) = I K I / V 2 7r r , a n a l o g o u s l y t o equation (2.7) must be equal to that given in equation (2.5). the homogeneous material case. The associated crack face That is displacements a distance r behind the tip are given by K2+K2l=q2KK (2.10) Ki-^/G.+a-i^/Gj] 1 of the elastic moduli. A further simplification is achieved when use is made of Dundurs' (1969) observation that for problems of this class the moduli dependence can be expressed in terms of just two (rather than three) special nondimensional combinations. In plane strain, Dundurs' parameters are
„
/2
a22 + ian^(Kl+iKn){2itr)^
1~"2
(2.7) -](*?+*?,) 2G, As discussed earlier, the remote stresses in the semi-infinite subsurface crack problem are required to approach (for all 6 but 6 = if) the characteristic Williams singular field of the interface crack, which can be written as
Journal of Applied Mechanics
G . d - ^ + G^l-K,)
(2.8)
(2.13)
and 1 G,(l-2^)-G2(l-2v1) 0 , ( 1 - ^ + 0,(1-^)
0= 2
(2.6)
where Kx and Ku are the standard Mode I and Mode II stress intensity factors. The energy release-rate is
Khk (2.18) feature of the solution for arbitrary small values of h/L. If the collection of terms <j> + e ln(h/L) is small, as might easily be the Note that AT, =Kl and Kn =K2 when a and (3 both vanish. ease judging from the systems listed in Table 2, then the stress intensity factors are well approximated by 3 Applications and Implications Moduli and values of a, /3, e, and / are presented for six representative material combinations in Table 2. The shear modulus and Poisson's ratio listed for each material are polycrystalline values derived from Simmons and Wang (1971). The values for the cubic materials are the average of 830/Vol. 54, DECEMBER 1987
Kj + iKn = qLkK =q\K\ eh
(3.6)
For example, in the case of the concentrated wedge force (3.1) Kl+iKll=q(P+iQ)(t!L/2Y'n
(3.7)
assuming e itself is small. Apart from the factor q, this is just Transactions of the ASME
Downloaded 21 May 2012 to 128.103.149.52. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
the classical result when no material discontinuity occurs. Similarly, the result for the finite crack (3.2) becomes
the other systems may occur for other geometries and loadings. The discussion and the analysis given above assume that 1/2 K, + iKu=q(o% + /(rr2)(7rZ,/2) (3.8) contact between the crack faces of the subinterface crack does which is again the classical result multiplied by q. In other not occur. In applications where the near tip conditions of the words, in these examples when e and <j> + e ln(h/L) are both subinterface crack is in pure Mode I and where h/L is not small, the ratio of the Mode II to Mode I stress intensity fac- ludicrously small, it is unlikely that contact of the crack faces tors is the same as in the corresponding classical problem but will be an issue. If the corresponding interface crack problem the energy release-rate is that of the interface crack as reflected does indicate contact well away from the tip, at distances as large as h or greater, then the possibility of contact in the by the factor q. subinterface crack problem should certainly be checked. SoluNow consider situations in fatigue, stress corrosion or under tions to (3.12) for h/L are only physically meaningful when monotonic loading when the crack will tend to advance in its h/L is not so small that contact will certainly occur or, what is own plane approximately parallel to the interface. Assuming more likely, that h is not so small that the crack lies so close to the fracture properties of material #2 are homogeneous along the interface that the material at the tip has properties which with its moduli, the crack will only advance in its plane if are affected by the existence of the interface. Kn=0. If Kn > 0 it will tend to deflect downward away from the interface, while if Kn < 0 it will tend to grow upward. By equation (3.5), the condition for the crack to advance parallel Acknowledgment to the interface in pure Mode I is This work was supported in part by the Office of Naval Research under Contract ONR-N00014-86-K-0753sin[7 + 0 + e/n(/)/L)] = 0 (3.9) Subagreement VB38639-0 from the University of California, or Santa Barbara, California, and by the Division of Applied y + <j> + eln(h/L)=2irn; « = 0,±1, ... (3.10) Sciences, Harvard University. with the associated Mode I intensity Kl=q\K\
References (3.11)
Values of h/L from equations (3.10) are h/L = exp[(2irn-y-)/e];
« = 0,±1, ...
(3.12)
but only those values (if any) will be physically meaningful which are small compared to unity but not so small that the parts of the crack faces make contact, as will be discussed below. The crack length L increases as the crack advances and thus h cannot remain strictly constant. However, if h at the tip satisfies equations (3.12) approximately as L increases the slope dh/dL of the crack, the path will be small (and equal to the value given by equations (3.12)), with the crack thus nearly paralleling the interface when h/L is small. As an illustration, consider the symmetric wedge loading (Q = 0) of the geometry in Fig. 2(a). By equations (3.1) and (3.3), 7 = 0. For the material systems listed in Table 2, the largest magnitude of e is 0.04, and it is readily seen that the only physically meaningful solution from equations (3.12), if any, is that associated with n = 0, i.e., h/L = exp[-<j>/e]
(3.13)
Of the systems in Table 2, only Cu/Si, Si/Cu, Ni/MgO, and MgO/Ni have positive values of $/e and might therefore propagate a subinterface crack of the kind envisioned here for this particular geometry and loading. For Cu/Si, h/L = 0.26; while for Si/Cu, h/L = 0.32. The accuracy of these estimates may be somewhat questionable since they probably lie outside the range of h/L where the asymptotic analysis is accurate. For Ni/MgO, h/L = 0.009 and for MgO/Ni, h/L = 0.011, and these estimates should be accurate. Evidently the crack could satisfy a Kn = 0 criterion by propagating near the interface in either phase. We do not investigate here the configurational stability of those paths but expect, following Cotterell and Rice (1980), that only a path with a negative crack-parallel nonsingular stress term at the tip is stable. The conclusions for a finite crack paralleling the interface in Fig. 2(b) under remote tensile loading ( ^ = 0) are similar. Now, 7 = 2e and h/L = exp[-2- is comparable.
Transactions of the ASME
Downloaded 21 May 2012 to 128.103.149.52. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
